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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="semi_af1.html#NM1">Semi_Affine_Space</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

















<b>assume </b><a NAME="E1:53"/><i><font color="Green">E44</font></i>: 
(  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 ;<br/>

<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_1"/>
not <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="semi_af1.html#R1" target="_self">is_collinear</a> 
 ;<br/><a NAME="E2:53_1"/><b>then </b>
(  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:53"><i><font color="Green">E44</font></i></a>, <a class="ref" href="semi_af1.html#T68" target="_self">Th68</a></i>;<br/><b>hence </b><a NAME="E3:53_1"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T67" target="_self">Th67</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_2"/>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="semi_af1.html#R1" target="_self">is_collinear</a> 
 ;<br/><a NAME="E2:53_2"/><b>then </b>
( not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:53"><i><font color="Green">E44</font></i></a>, <a class="ref" href="semi_af1.html#T61" target="_self">Th61</a></i>;<br/><a NAME="E3:53_2"/><b>then </b>
(  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T68" target="_self">Th68</a></i>;<br/><b>hence </b><a NAME="E4:53_2"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T67" target="_self">Th67</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:53_3"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  )
 ;<br/><a NAME="E2:53_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:53"><i><font color="Green">E44</font></i></a>, <a class="ref" href="semi_af1.html#T54" target="_self">Th54</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:53_3"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="semi_af1.html#T66" target="_self">Th66</a></i>;<br/><a NAME="E5:53_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E4:53_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="semi_af1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E6:53_3"/><i><font color="Green">E50</font></i>: 
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="semi_af1.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:53"><i><font color="Green">E44</font></i></a>, <a class="ref" href="semi_af1.html#T56" target="_self">Th56</a></i>;<br/><a NAME="E7:53_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T12" target="_self">Th12</a></i>;<br/><a NAME="E8:53_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:53"><i><font color="Green">E44</font></i></a>, <a class="ref" href="semi_af1.html#T54" target="_self">Th54</a></i>;<br/><a NAME="E9:53_3"/><b>then </b>
not <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="semi_af1.html#R1" target="_self">is_collinear</a> 
 <b>by </b><i><a class="txt" href="semi_af1.html#E4:53_3"><i><font color="Green">E48</font></i></a>, <a class="txt" href="semi_af1.html#E5:53_3"><i><font color="Green">E49</font></i></a>, <a class="txt" href="semi_af1.html#E6:53_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="semi_af1.html#E7:53_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="semi_af1.html#T39" target="_self">Th39</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E11:53_3"/><i><font color="Green">E52</font></i>: 
 <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T62" target="_self">Th62</a></i>;<br/><a NAME="E12:53_3"/><i><font color="Green">E53</font></i>: 
 <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:53"><i><font color="Green">E44</font></i></a>, <a class="txt" href="semi_af1.html#E4:53_3"><i><font color="Green">E48</font></i></a>, <a class="txt" href="semi_af1.html#E11:53_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="semi_af1.html#T69" target="_self">Th69</a></i>;<br/><a NAME="E13:53_3"/><b>then </b>
( not <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  &amp; not <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="semi_af1.html#R1" target="_self">is_collinear</a>  )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:53_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="semi_af1.html#E11:53_3"><i><font color="Green">E52</font></i></a>, <a class="ref" href="semi_af1.html#T57" target="_self">Th57</a></i>;<br/><a NAME="E14:53_3"/><b>then </b>
(  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp;  <a href="semi_af1.html#R2" target="_self">parallelogram</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="semi_af1.html#E1:53"><i><font color="Green">E44</font></i></a>, <a class="txt" href="semi_af1.html#E11:53_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="semi_af1.html#E12:53_3"><i><font color="Green">E53</font></i></a>, <a class="ref" href="semi_af1.html#T68" target="_self">Th68</a></i>;<br/><b>hence </b><a NAME="E15:53_3"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="semi_af1.html#T67" target="_self">Th67</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:53"/>
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="semi_af1.html#E2:53"><i><font color="Green">E45</font></i></a>, <a class="txt" href="semi_af1.html#E3:53"><i><font color="Green">E46</font></i></a></i>;<br/>


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